$ 1. \( y(0) = 2 \): \[ C_1 + C_2 = 2 \] 2. \( y'(t) = \frac{\sqrt{15}}{5} C_1 e^{\frac{\sqrt{15}}{5} t} - \frac{\sqrt{15}}{5} C_2 e^{-\frac{\sqrt{15}}{5} t} \] Plugging \(t = 0\): \[ \frac{\sqrt{15}}{5}(C_1 - C_2) = 0 \] \[ C_1 = C_2 \] Substituting \(C_1 = C_2\) into the first condition: \[ 2C_1 = 2 \] \[ C_1 = 1 \quad \text{and} \quad C_2 = 1 \] Thus, the solution is: $$ \[ y(t) = e^{\frac{\sqrt{15}}{5} t} + e^{-\frac{\sqrt{15}}{5} t} \] $